OFFSET
0,1
REFERENCES
Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
LINKS
Karim Belabas and Henri Cohen, Computation of sum_{p prime} 1/(p^s log(p)), PARI/GP script, 2020.
Henri Cohen, High-precision calculation of Hardy-Littlewood constants, (1998).
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 [math.NT], 2009-2018, Section 2.4.
EXAMPLE
0.50778218785919931877437510379470557...
MATHEMATICA
digits = 106; precision = digits + 15;
tmax = 400; (* integrand considered negligible beyond tmax *)
kmax = 400; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[Log[Zeta[t]], {t, k, tmax},
WorkingPrecision -> precision, MaxRecursion -> 20,
AccuracyGoal -> precision];
f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^2)*InLogZeta[2k]]];
s = 0; Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
RealDigits[s][[1]][[1 ;; digits]] (* Jean-François Alcover, Feb 06 2021, updated Jun 23 2022 *)
PROG
(PARI) \\ See Belabas, Cohen link. Run as SumEulerlog(2) after setting the required precision.
(PARI) default(realprecision, 200); s=0; for(k=1, 300, s = s + moebius(k)/k^2 * intnum(x=2*k, [[1], 1], log(zeta(x))); print(s)); \\ Vaclav Kotesovec, Jun 12 2022
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
N. J. A. Sloane, Jan 26 2013
EXTENSIONS
More terms from Hugo Pfoertner, Feb 01 2020
More digits from Vaclav Kotesovec, Jun 12 2022
STATUS
approved